Dirac Exchange Research Articles

Two-Term Asymptotics of the Exchange Energy of the Electron Gas on Symmetric Polytopes in the High-Density Limit

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations

We derive the next order correction to the Dirac exchange energy for the free electron gas in a box with zero boundary conditions in the thermodynamic limit. The correction is of the order of the surface area of the box, and comes from three different contributions: (i) a real-space boundary layer, (ii) a boundary-condition-induced small shift of Fermi momentum and bulk density, and (iii) a long-range electrostatic finite-size correction. Moreover we show that the local density approximation, in addition to capturing the bulk term exactly, also produces a correction of the correct order but not the correct size. Generalized gradient approximation (GGA) corrections are found to be capable of capturing the surface term exactly, provided the gradient enhancement factor satisfies a simple explicit integral constraint. For current GGAs such as B88 and Perdew, Burke and Ernzerhof we find that the new constraint is not satisfied and the size of the surface correction is overestimated by about ten percent. The new constraint might thus be of interest for the design of future exchange functionals.

Derivation of dirac exchange interaction potential from quantum plasma kinetic theory

The Dirac exchange interaction is derived from recent quantum kinetic theory for collisionless plasmas. For this purpose, the kinetic equation is written in the semiclassical and long wavelength approximations. The validity of the model for real systems is worked out, in terms of temperature and density parameters. Within the region of applicability, the correlation potential energy is shown to be always smaller than the exchange contribution. From the moments of the quantum kinetic equations, macroscopic, hydrodynamic equations are found, for an electron-ion plasma. The Dirac exchange term is explicitly derived, in the case of a completely degenerate electron gas. These results show, within quantum kinetic theory for charged particle systems, a new view of the Dirac exchange interaction frequently used in density functional theory parametrization. Finally, a simpler form of the quantum plasma exchange kinetic theory is also found.

Gaussian basis functions for an orbital‐free‐related density functional theory of atoms

AbstractA representation of polymer self‐consistent field theory equivalent to quantum density functional theory is given in terms of non‐orthogonal basis sets. Molecular integrals and self‐consistent equations for spherically symmetric systems using Gaussian basis functions are given, and the binding energies and radial electron densities of neutral atoms hydrogen through krypton are calculated. An exact electron self‐interaction correction is adopted and the Pauli‐exclusion principle is enforced through ideas of polymer excluded‐volume. The atoms hydrogen through neon are examined without some approximations which permit cancellation of errors and spontaneous shell structure is observed. Correlations are neglected in the interest of simplicity and comparisons are made with Hartree–Fock theory. The implications of the Pauli‐exclusion potential and its approximate form are discussed, and the Pauli model is analyzed using scaling theory for the uniform electron density case where the correct form of the Thomas–Fermi quantum kinetic energy and the Dirac exchange correction are recovered.

Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

Dissociation limit in Kohn–Sham density functional theory

We consider the dissociation limit for molecules of the type X2 in the Kohn–Sham density functional theory setting, where X can be any element with N electrons. We prove that when the two atoms in the system are torn infinitely far apart, the energy of the system converges to minα∈[0,N](IαX+I2N−αX), where IαX denotes the energy of the atom with α electrons surrounding it. Depending on the “strength” of the exchange this minimum might not be equal to the symmetric splitting 2INX. We show numerically that for the H2-molecule with Dirac exchange this gives the expected result of twice the energy of a H-atom 2I1H.

The Nonlinear Schrödinger Equation for Orthonormal Functions: Existence of Ground States

We study the nonlinear Schr\"odinger equation for systems of $N$ orthonormal functions. We prove the existence of ground states for all $N$ when the exponent $p$ of the non linearity is not too large, and for an infinite sequence $N_j$ tending to infinity in the whole range of possible $p$'s, in dimensions $d\geq1$. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant.

Pair Densities in Density Functional Theory

The exact interaction energy of a many-electron system is determined by the electron pair density, which is not well-approximated in standard Kohn--Sham density functional models. Here we study the (complicated but well-defined) exact universal map from density to pair density. We survey how many common functionals, including the most basic version of the local density approximation (Dirac exchange with no correlation contribution), arise from particular approximations of this map. We develop an algorithm to compute the map numerically and apply it to one-parameter families $\{\alpha\rho(\alpha x)\}_{\alpha>0}$ of one-dimensional homogeneous and inhomogeneous single-particle densities. We observe that the pair density develops remarkable multiscale patterns which strongly depend on both the particle number and the “width” $\alpha^{-1}$ of the single-particle density. The simulation results are confirmed by rigorous asymptotic results in the limiting regimes $\alpha\gg$ and $\alpha\ll1$. For one-dimensional homogeneous systems, we show that the whole spectrum of patterns is reproduced surprisingly well by a simple asymptotics-based ansatz which slowly smoothens out the “strictly correlated” $\alpha=0$ pair density while slowly turning on the $\alpha=\infty$ “exchange” terms as $\alpha$ increases. Our findings lend theoretical support to the celebrated semi-empirical idea [A. D. Becke, J. Chem. Phys., 98 (1993), pp. 5648--5652] to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system-independent constant.

Modified Slater exchange potential with correct uniform electron gas limit

Bulat and Levy recently deduced an expansion for the exact-exchange potential whose leading term is Slater’s averaged exchange-charge potential modified through the inclusion of orbital eigenvalues. When used as a stand-alone approximation to the exact-exchange potential, this leading term is much more accurate than Slater’s original. We give an alternative derivation of this term and show that, unlike the Slater potential, it correctly reduces to the Dirac exchange potential (i.e., the local density approximation) for a uniform electron gas, provided that the zero-energy reference of orbital eigenvalues is chosen as the Fermi level of the system. The orbital-energy-modified Slater potential should also recover the Dirac exchange potential for neutral atoms in the limit of infinite nuclear charge, but in practice converges to this limit very slowly on account of extreme sensitivity to the choice of zero-energy reference around the Fermi level.

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others.

Testing the Majorana nature of gluinos and neutralinos

Gluinos and neutralinos, supersymmetric partners of gluons and neutral electroweak gauge and Higgs bosons, are Majorana particles in the minimal supersymmetric standard model (MSSM). Decays of such self-conjugate particles generate charge symmetric ensembles of final states. Moreover, production channels of supersymmetric particles at colliders are characteristically affected by the Majorana nature of particles exchanged in the production processes. The sensitivity to the Majorana character of the particles can be quantified by comparing the predictions with Dirac exchange mechanisms. A consistent framework for introducing gluino and neutralino Dirac fields can be designed by extending the N=1 supersymmetry of the MSSM to N=2 in the gauge sector. We examine to which extent like-sign dilepton production in the processes qq{yields}q-tildeq-tilde and e{sup -}e{sup -}{yields}e-tilde{sup -}e-tilde{sup -} is affected by the exchange of either Majorana or Dirac gluinos and neutralinos, respectively, at the Large Hadron Collider (LHC) and in the prospective e{sup -}e{sup -} mode of a lepton linear collider.

Ionization potentials of neutral atoms and positive ions in the limit of large atomic number

The many-electron Schr\"odinger equation allows one to make predictions for the nonrelativistic ionization potentials of both highly charged positive atomic ions and neutral atoms in the limit of large atomic number $Z$. Beginning with theoretical configuration interaction data on both Li- and Be-like positive atomic ions by K.T. Chung et al. [Phys. Rev. A 45, 7766 (1992); Phys. Rev. A 47, 1740 (1993)], their nonrelativistic first ionization potentials $I$ are plotted in the form of $I∕{Z}^{2}$ versus $1∕Z$. In these two sequences, it is then clear that ${\mathrm{lim}}_{Z\ensuremath{\rightarrow}\ensuremath{\infty}}(I∕{Z}^{2})$ remains finite and that this limit is approached essentially linearly. Interpretation is afforded by means of the so-called $1∕Z$ expansion of atomic theory. The question of the behavior of the nonrelativistic ionization potentials for large $Z$ is then addressed for neutral atoms. We perform density functional theory (DFT) calculations using the local density approximation with Dirac exchange and neglecting Coulomb correlation for the heavy actinides as well as for the rare gas and alkali atoms from $Z=10$ up to $Z=291$. Our results suggest that there is a nonzero value for ${\mathrm{lim}}_{Z\ensuremath{\rightarrow}\ensuremath{\infty}}I$ for both the noble gases and the alkalis, the second one implying that there is a lower bound of about $2.56\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ to the ionization potential of any nonrelativistic neutral atom. We also explore an alternative approach via the one-body potential $V(r)$ of DFT by calculating the eigenenergies of the highest occupied atomic orbital in neutral atoms. Finally, relativistic corrections needed to compare measured ionization potentials to the predictions from the Schr\"odinger equation are briefly considered.

Next generation of ab initio energy density functionals: Thoughts and facts from solvable models

AbstractAfter a brief summary of trends revealed in the presentations of this DFT Conference, especially on ab initio energy density functionals and potentials, some emphasis will be given to the likely “next generation” of first‐principles energy density functionals. The Slater exchange potential V(r) then seems a natural starting point, being related to the Dirac exchange energy density εx(r) and the ground‐state density n(r) by V(r) = 2εx(r)/n(r). We then discuss various “corrections” to this Slater potential V(r). Although usually small compared with the exchange potential, the correlation contribution Vc(r) remains difficult to treat from first principles, but we refer to a few directions stemming from (generally two‐electron) solvable models. Finally, we give some attention to the kinetic energy, especially the single‐particle functional Ts[n] but also, again though from simple examples, to the correlation kinetic energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

The optimal size of the exchange hole and reduction to one-particle Hamiltonians

We use the concept of the exchange hole introduced by Slater to bound the energy of atoms, molecules, and other systems interacting by Coulomb forces from below by one-particle Hamiltonians with an effective screening potential and an exchange hole around each electron. Interestingly enough the optimal size of the exchange hole is smaller than Slater proposed: the best lower bound is obtained when the exchange hole carries charge 1/2 instead of 1. To highlight the quality of our estimate we show that the Dirac exchange energy with a slightly different constant bounds the exchange–correlation energy from below, an estimate previously derived by Lieb and later improved by Lieb and Oxford.

Full potential calculations on the electron bandstructures of Sphalerite, Pyrite and Chalcopyrite

The bulk electronic structures of Sphalerite, Pyrite and Chalcopyrite have been calculated within an ab initio, full potential, density functional approach. The exchange term was approximated with the Dirac exchange functional, the Vosko–Wilk–Nusair parameterization of the Cepler–Alder free electron gas was used for correlation and linear combinations of Gaussian type orbitals were used as basis functions. The Sphalerite (zinc blende) band gap was calculated to be direct with a width of 2.23 eV. The Sphalerite valence band was 5.2 eV wide and composed of a mixture of sulfur and zinc orbitals. The band below the valence band located around −6.2 eV was mainly composed of Zn 3d orbitals. The S 3s orbitals gave rise to a band located around −12.3 eV. Pyrite was calculated to be a semiconductor with an indirect band gap of 0.51 eV, and a direct gap of 0.55 eV. The valence band was 1.25 eV wide and mainly composed of non-bonding Fe 3d orbitals. The band below the valence band was 4.9 eV wide and composed of a mixture of sulfur and iron orbitals. Due to the short inter-atomic distance between the sulfur dumbbells, the S 3s orbitals in Pyrite were split into a bonding and an anti-bonding range. Chalcopyrite was predicted to be a conductor, with no band-crossings at the Fermi level. The bands at −13.2 eV originate from the sulfur 3s orbitals and were quite similar to the sulfur 3s bands in Sphalerite, though somewhat shifted to lower energy. The top of the valence band consisted of a mixture of orbitals from all the atoms. The lower part of the same band showed metal character. Computational modeling as a tool for illuminating the flotation and leaching processes of Pyrite and Chalcopyrite, in connection with surface science experiments, is discussed.

The partitioning of Thomas–Fermi, von Weizsäcker and Dirac density functionals

The partitioning of several classical density functionals, including the Thomas–Fermi (TF), Weizsacker kinetic energies and the Dirac exchange energy, are studied for the partitioned systems that satisfy the orthogonality condition. The partitioning of the TF kinetic energy shows a raising of the energy of the valence electrons which is not found for the partitioning of the Weizsacker kinetic energy. The partitioning of the Weizsacker kinetic energy is not unique and contains a correlation part. The partitioning of the Dirac exchange energy shows that the exchange energy of a subsystem can be largely influenced by the other subsystems when this subsystem does not correspond to the lowest energy subspace. The self-interaction calculated from the partitioned exchange energies is different from previous results.

Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set

Scaling factors for obtaining fundamental vibrational frequencies from harmonic frequencies calculated at six of the most commonly used levels of theory have been determined from regression analysis for the polarized-valence triple-zeta (pVTZ) Sadlej electric property basis set. The Sadlej harmonic frequency scaling factors for first- and second-row molecules were derived from a comparison of a total of 900 individual vibrations for 111 molecules with available experimental frequencies. Overall, the best performers were the hybrid density functional theory (DFT) methods, Becke's three-parameter exchange functional with the Lee–Yang–Parr fit for the correlation functional (B3-LYP) and Becke's three-parameter exchange functional with Perdew and Wang's gradient-corrected correlation functional (B3-PW91). The uniform scaling factors for use with the Sadlej pVTZ basis set are 0.9066, 0.9946, 1.0047, 0.9726, 0.9674 and 0.9649 for Hartree–Fock, the Slater–Dirac exchange functional with the Vosko–Wilk–Nusair fit for the correlation functional (S-VWN), Becke's gradient-corrected exchange functional with the Lee–Yang–Parr fit for the correlation functional (B-LYP), B3-LYP, B3-PW91 and second-order Moller–Plesset theory with frozen core (MP2(fc)), respectively. In addition to uniform frequency scaling factors, dual scaling factors were determined to improve the agreement between computed and observed frequencies. The scaling factors for the wavenumber regions below 1800 cm−1 and above 1800 cm−1 are 0.8981 and 0.9097, 1.0216 and 0.9857, 1.0352 and 0.9948, 0.9927 and 0.9659, 0.9873 and 0.9607, 0.9844 and 0.9584 for Hartree–Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2(fc), respectively. Hybrid DFT methods along with the Sadlej pVTZ basis set provides reliable theoretical vibrational spectra in a cost-effective manner.

Left-right correlation energy

We first attempt to determine a local exchange functional Ex[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. Ex[p is determined from p and |δp|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ i |2 improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local Ex[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.

Spin-triplet superconducting pairing due to local Hund’s rule and Dirac exchange

We discuss general implications of the local spin-triplet pairing among fermions induced by local ferromagnetic exchange, example of which is the Hund's rule coupling. The quasiparticle energy and their wave function are determined for the three principal phases with the gap, which is momentum independent. We utilize the Bogolyubov-Nambu-De Gennes approach, which in the case of triplet pairing in the two-band case leads to the four-components wave function. Both gapless modes and those with an isotropic gap appear in the quasiparticle spectrum. A striking analogy with the Dirac equation is briefly explored. This type of pairing is relevant to relativistic fermions as well, since it reflects the fundamental discrete symmetry-particle interchange. A comparison with the local interband spin-singlet pairing is also made.

Nonrelativistic exchange-energy density and exchange potential in the lowest order of the1/Zexpansion for ten-electron atomic ions

The Dirac exchange energy ${E}_{x}$ has a density ${\ensuremath{\varepsilon}}_{x}(r)$ in a spherically symmetric ten-electron atomic ion that is determined by the idempotent first-order density matrix (1 DM). In turn, this 1 DM has as its leading term in the $1/Z$ expansion, with Z the atomic number, the bare Coulomb result. This latter quantity is known analytically from the study of March and Santamaria [Phys. Rev. A 38, 5002 (1988)]. Here ${\ensuremath{\varepsilon}}_{x}(r)$ is calculated analytically, and presented graphically for $Z=92$ in this large-Z limit. The Slater approximation to the exchange potential ${V}_{x}(\mathbf{r}),$ namely, ${V}_{\mathrm{Sla}}(\mathbf{r})=2{\ensuremath{\varepsilon}}_{x}(r)/\ensuremath{\rho}(r)$ with $\ensuremath{\rho}(r)$ the ground-state electron density, is also plotted for $Z=92.$ In the large-Z limit, ${V}_{x}(\mathbf{r})$ can be obtained by functional differentiation of the resultant exchange energy, and is expressed in terms of electron and kinetic-energy densities. Numerical calculations, based on ${V}_{\mathrm{Sla}}(\mathbf{r})$ plus approximate corrections, are also presented.